Non-Hermitian topology and criticality in photonic arrays with engineered losses

Integrated photonic systems provide a flexible platform where artificial lattices can be engineered in a reconfigurable fashion. Here, we show that one-dimensional photonic arrays with engineered losses allow the realization of topological excitations stemming from non-Hermiticity and bulk mode criticality. We show that a generalized modulation of the local photonic losses allows the creation of topological modes both in the presence of periodicity and even in the quasiperiodic regime. We demonstrate that a localization transition of all the bulk photonic modes can be engineered in the presence of a quasiperiodic loss modulation, and we further demonstrate that such a transition can be created in the presence of both resonance frequency modulation and loss modulation. We finally address the robustness of this phenomenology to the presence of next to the nearest neighbor couplings and disorder in the emergence of criticality and topological modes. Our results put forward a strategy to engineer topology and criticality solely from engineered losses in a photonic system, establishing a potential platform to study the impact of nonlinearities in topological and critical photonic matter.

In this manuscript, we present a strategy to engineer topological modes and criticality simultaneously in one-dimensional photonic arrays solely based on engineered losses.In particular, we show that a generalized set of models with engineered losses feature topological edge modes stemming from non-Hermitian topology.For quasiperiodic modulations, we show that the modulated losses lead to a delocalization to localization transition of the bulk states.We analyze the resilience of the topological edge modes to disorder in the engineered losses and detuning frequency, and the impact of long-range tunneling in the localization transition and topological modes.Our results provide a starting point for design-ing topological photonic devices based on tunable losses.Our manuscript is organized as follows.In section II, we present the generalized model featuring modes from engineered losses.In section III, we analyze the localization transition driven by modulated losses.In Sec.IV, we address the impact of perturbations and disorder.In Sec.V we address the continuum limit of the model.Finally, in Sec.VI, we summarize our conclusions.

II. TOPOLOGICAL MODES FROM ENGINEERED LOSSES
We consider a one-dimensional array of photonic dots featuring localized excitation.Photonic losses are included by adding a non-Hermitian term into a 1D model in each site of the array.For the sake of concreteness, we first consider an engineered loss with four-site periodicity [36,37,[50][51][52][53], as shown in Fig. [1(a)], whose Hamiltonian takes the form where a † n creates photon in site n, v 0 + v I β n , with v 0 , v I real numbers denotes the site-dependent loss, parametrised by the modulation (β 1 = β 4 = 1) and (β 2 = β 3 = −1).The term v 0 leads to an overall loss in the system, and therefore in the following it will be factored out in the spectra.Due to the non-Hermiticity of the Hamiltonian, the eigenenergies E α will be in general complex, with H|Ψ α ⟩ = E α |Ψ α ⟩.We show in In particular, as loss modulation strength increases the edge modes get localized at the end sites as compared to those in the bulk.
The previous topological non-Hermitian model given by Eq. [1] can be seen as a specific case of a generalized non-Hermitian model given by the following Hamiltonian where α is the inverse period of modulation, φ is the phase of modulation, and v I is the amplitude of modulation of the losses.This model can be realized as the non-Hermitian generalization of the Aubry-André-Harper (AAH) potential, [20].In its Hermitian form v sin(2πnα + φ)a † n a n , this model is well known to be equivalent to a twodimensional quantum Hall system, [22], thus inheriting topological edge modes.However, such a mapping cannot be performed in its non-Hermitian generalization.As it is shown in Fig. [1(a)], the systems in Eq. [1] and in Eq. [2] are equivalent when α = 1  4 and φ = 3π 4 .To have a periodic system, the potential should be commensurate to the lattice periodicity.In particular, when α is a rational number p/q, the total number of sites, N should be a multiple of q.While for a quasiperiodic system, α should be an irrational number.For different values of α and φ, we obtain a series of topologically inequivalent insulators [54].
We now show how topological modes can appear for the generic values of the parameters α and φ [54,55].We show in Fig. [2(a-d)], the real part of the energy spectrum for different values of the parameter α.In the case of α = 1/4 (Fig. [2(a)]), the topological edge modes emerge at zero energy.In contrast, topological modes at other values of α appear at finite energies.Also, the real part of the bulk modes is symmetric with respect to the band gap at Re(E) = 0 which is a feature of particle-hole symmetry.The particle-hole symmetry is associated with the system when q is a multiple of 4. Figs.[2(a),(b)] show the commensurate limit [54], where the frequency of the modulation leads to a periodic system.Figs.[2(c),(d)] are for incommensurate frequencies, we can see that the energy spectra have a fractal nature for these cases and have edge modes shown in red.Also, many band gaps in the energy spectrum do not have robust edge states, as can be inferred from the figure.Note, all these systems shown in Fig. [2(a-d)] an have imaginary component of the energy that is not shown in the figure.
We can also study the presence of edge modes for systems with a range in α by computing the spectra of the system as a function of the modulation frequency α.In its Hermitian version, such a plot is known as the Hofstadter butterfly spectrum of the Hamiltonian.We show the real part of the Hofstadter spectrum of Hamiltonian H given by Eq. [2] in Fig. [2(e)].The edge modes are plotted in red which appear in band gaps for various α, showing the appearance of those modes even for modulation frequencies not commensurate with the lattice.

III. CRITICALITY AND LOCALIZATION-DELOCALIZATION TRANSITION
Quasiperiodic Hermitian models feature localization transitions at finite strength, phenomena that turned them into an attractive platform to realize wavefunction criticality [27,[56][57][58][59][60].The Hermitian AAH model described by an onsite potential v sin(2παn + φ) is known to have a localization transition as a function of the modulation strength v ∈ R.This model is self-dual and has a limit of self-duality at v = 2t, i.e. all the bulk states localize at v = 2t [20].The localization transition as a function of v is independent of the phase of modulation φ ∈ R for a quasiperiodic system [20].In the following, we study the localization transition of the non-Hermitian AAH model.The localization transition can be directly inferred from the calculation of the inverse participation ratio (IPR) of the eigenstates [49].For a state |ψ⟩, the IPR is defined as For N → ∞, we have IPR = 0 for an extended state and IPR ∼ 1/W , with W the number of sites where the state is localized, for a localized state.We study this transition for the system described by Eq. [2] with respect to the modulation strength v I .We show in Fig. [3(a)] and Fig. [3(b)] the IPR for all the eigenvalues of the Hamiltonian given by Eq. [2] with respect to the amplitude of modulation v I for α = ( √ 5 − 1)/2.We can see that a localization transition for all the eigenstates occurs at v I = 2t, simultaneously for all eigenstates.As a reference, the maximum value of IPR in the figures is of the order 18/N .In the Hermitian version of this model, a similar phenomenology takes place stemming from self-duality between the coordinate and the momentum space.It should be contrasted that in the case of conventional disorder, the localization transition occurs at infinitesimally small disorder for a onedimensional model [61,62].The existence of a critical value directly reflects the inherent quasiperiodicity of the potential, making this model genuinely different from a disordered system.
The existence of a localization transition in a non-Hermitian model, analogous to the one known in its Hermitian counterpart, motivates the question of whether there exists a generalized model featuring such a localization-delocalization.For this purpose, we now address the localization transition for a complex modulation strength.Consider the following modification for our Hamiltonian, where v I is the modulation of the loss and v R the modulation of the onsite potential of the system, such that v R , v I are real parameters.In the case v I = 0, the previous model is equivalent to the Hermitian AAH model, whereas for v R = 0 we recover our model with modulated losses.We start by fixing v R as a nonzero value to study the spectrum as a function of the loss modulation v I .We show in Fig.The localization-delocalization transition can also be studied from the localization length of the wavefunction.In the localized limit, localized eigenstates can be fitted to a functional form such as where α labels the eigenstate, n is the site of the chain and r 0 is the center of eigenstate i.e.where |ψ α | 2 (n) is the highest.The parameter 1/λ is the localization length, which in the case of an extended state corresponds to 1/λ = ∞.For each eigenstate of the system, we perform a fit to the previous functional form (Eq. [5]), which allows extracting a localization length for each state.We show in Fig.

IV. LONG-RANGE COUPLING AND DISORDER
A. Impact of long-range coupling So far our analysis has focused on the limit featuring first nearest-neighbor coupling.In realistic experimental scenarios, finite coupling between longer neighbors may occur.Couplings beyond first neighbors are expected to give rise to an energy-dependent localization transition.In particular, the Hermitian AAH model with an exponentially decreasing hopping te −p|n−n ′ | a † n a n ′ gives rise to an energy-dependent localization transition that can be derived analytically [63].The non-Hermitian limit however cannot be addressed with the self-duality procedure of the Hermitian limit, and thus we will focus here on an exact numerical strategy.For this purpose, we now address the impact of second and third-nearest-neighbor coupling.The Hamiltonian for this system takes the form We show in Fig. [5], the real part of the energy spectrum for the Hamiltonian given by Eq. [6] with respect to the modulation strength v I when v R = 0. Panels Fig. [5(a)] and Fig. [5(b)] show the evolution of the localization of the bulk modes as a function of v I in the presence of second nearest neighbor hopping t 2 .It is observed that the inclusion of second neighbor hopping removes the particle-hole symmetry of the spectra.Interestingly, we observe that at certain energies such as v I = 1.9t, localized and extended states coexist in the bulk.This must be contrasted with the situation observed in Fig. [3] where it was observed that all the eigenstates are either extended or localized, and no coexistence is possible.The coexistence of localized and extended states is associated with a mobility edge, and in Fig. [5(a,b)] we observe that this mobility edge depends on the strength t 2 , appearing in a wider region for increasing t 2 .It is instructive to address another case with extended hopping, in particular, third neighbor hopping t 3 as shown in Fig. [5(c,d)], while taking t 2 = 0.In this scenario, we observe that the spectrum remains particle-hole symmetric to Re(E) = 0.This extended model also features a localization transition as a function of v I , happening at different parameter values depending on the energy.In particular, the bulk states with energy |Re(E)| closer to 0 get localized for a smaller v I as compared to those that are farther.Similar phenomenology is observed in a fully Hermitian model, where mobility edge exists for the second and third neighbour hopping.These results highlight that higher-order couplings lead to an energy-dependent localization of the non-Hermitian bulk states, regardless of whether they maintain the particle-hole symmetry of the underlying model.

B. Impact of disorder
We now study the robustness of the system given by Eq. [2] as a function of disorder in the imaginary onsite energy [54].For the sake of concreteness, we will focus on the model featuring modulation solely on the losses by taking v R = 0 and v I is nonzero corresponding to the model featuring modes at zero energy for α = 1/4.To study the effect of disorder on the energy spectrum, we define the spatially resolved spectral density as where ω is the frequency.We note that the previous quantity projects onto the real part of the eigenenergy, whereas an analogous one can be defined for the imaginary part.The spectral density D(ω) provides direct access to the number of eigenstates with a specific value in the real part of the energy, and is analogous to the density of states in a Hermitian system.
We study the spectral density for the Hamiltonian given by Eq. [2] with respect to the disorder strength, considering loss disorder and detuning disorder.The disorder is included in the Hamiltonian by including a term where w I and w R parametrize the loss and detuning disorder, respectively.The disorder is included by sampling a Gaussian distribution χ with an average value 0 and width 1, and for the sake of simplicity we will consider loss and detuning disorder separately.Let us first focus on the disorder in the loss modulation.We show in ).This phenomenology highlights that the topological zero mode is robust to the presence of disorder in the loss.We now move on to the situation where disorder appears in the resonance frequency of each site (Fig. ).However, it is observed that at the edge the energy of the zero modes is no longer pinned at zero energy, leading to an edge state at a finite energy proportional to the typical level of the disorder ((Fig.[6(d)])).This phenomenology highlights that while disorder in the loss does not impact the resonance frequency of the edge mode, disorder in the onsite energies affects the topological edge mode.

V. THE AUBRY-ANDR É-HARPER MODEL IN THE CONTINUUM LIMIT
Previously we have focused on the discrete AAH model.In the following, we now bring our attention to the non-Hermitian AAH model in the continuum limit.The continuous AAH model is given by the Hamiltonian in continuous space as where p = −i ∂ ∂x , m the effective mass, α is the AAH frequency and Ψ † x , Ψ x continuum field operators fulfilling [Ψ x , Ψ † x ′ ] = δ(x − x ′ ).The eigenbasis of the previous Hamiltonian can be computed by solving the associated Sturm-Liouville non-Hermitian differential equation of the form − , with ϵ k the complex eigenvalue and k parametrizing the phase picked due to twisted boundary conditions ψ k (x + 1/α) = e ik ψ k (x).It is worth noting that in this continuum limit, the absence of an underlying lattice makes the Hamiltonian explicitly periodic in space, with a periodicity 1/α.
In the absence of a potential v R = v I = 0, the photonic dispersion ϵ(k) corresponds to the folded band structure of a free particle p 2 /(2m) as shown in Fig. 7(a).Due to the periodicity of the potential 1/α, the quasimomentum k must be unfolded to the original free momentum p to recover a parabolic dispersion even for v R = v I = 0.For the sake of comparison with the free particle limit, in the following, we will perform an unfolding of the eigenvalues as a function of the momentum k in the unit cell of size 1/α, which in the case v R = v I = 0 gives rise to the original band dispersion ϵ(p) of Fig. 7(b) for the unfolded momentum p.With the previous methodology to solve the continuum model and unfold its eigenvalues we first address the Hermitian AAH model, and later move to the non-Hermitian version.Focusing first on the Hermitian case v R ̸ = 0 and v I = 0, we show in Fig. [7(c,d)] the unfolded eigenvalues for two strengths of the AAH potential.The insets of Fig. [7(c,d)] show the photonic dispersion in the original quasimomentum space k before the unfolding is performed.It is observed that at the lowest energies, weakly dispersive states appear, giving rise to a set of minibands with weak dispersion, that eventually lead to a highly dispersive state at high energies recovering the free particle dispersion.The emergence of those minibands is easily rationalized from the fact that, at strong v R , the Hamiltonian describes a set of deep harmonic potentials, each one leading to harmonic oscillator modes.Due to the finite depth of the potential, harmonic oscillator modes between different potential wells have a finite overlap, leading to a weak dispersion of the individual modes.At higher energies, equivalent to higher modes of the oscillator, the tunneling between different wells becomes stronger.Finally, at kinetic energies bigger than the depth of the well, the eigenstates resemble the free particle dispersion.We now move on to the non-Hermitian AAH continuous model with v R = 0 and v I ̸ = 0, as shown in Fig. [7(e,f)].In this limit, it is observed that at low energies a set of weakly dispersing modes appear leading to a set of nearly flat mini-bands.At higher energies, the free particle dispersion is recovered, analogously to the Hermitian case.It is further observed that at high energies a small replica of the dispersion is obtained due to the momentum scattering created by the non-Hermitian potential.In contrast with the Hermitian case, the emergence of weakly dispersive states is no longer associated with harmonic oscillator modes in each well, but they stem purely from the non-Hermitian potential.
The nature of the states in Fig. [7] can be further elucidated by computing the spatially resolved continuum spectral density that takes the form The spatially resolved spectral density for the continuum Hermitian and non-Hermitian AAH models are shown in Fig. (v I = 0)], it is observed that the low energy states are localized at the bottom of the Hermitian potential wells, as expected from harmonic oscillator modes.As the energy is increased, the spatial extension of the modes becomes bigger, leading to an increasing overlap between the states in the different wells, and accounting for the enhancement of the bandwidth shown in Fig. [7(c,d)].At high energies, bigger than the top of the potential, nearly free wavefunctions are recovered extending through all x.We now move on to consider the modes of the non-Hermitian potential (v R = 0), shown in Fig. [8(b)].In the non-Hermitian case, it is observed that the lowest energy modes are located both at the maxima and minima of the non-Hermitian potential.At higher frequencies, the extension of the confined modes becomes bigger, leading to the increased bandwidth observed in Fig. [7(e,f)].Finally, at high enough energies, the nearly free particle dispersion is recovered, with the unique phenomenology that the states remain peaked in specific regions of the non-Hermitian potential, in particular those with a vanishing value.This phenomenology stems from the nonperturbative nature of the non-Hermitian potential, in stark contrast with the Hermitian case.

VI. CONCLUSION
We have shown how tunable local losses allow engineering topological modes in a photonic system.In particular, we showed that both periodically engineered and quasiperiodic loss profiles allow the creation of topological excitations at the edge of the photonic array.In the quasiperiodic limit, we showed that a critical localization takes place both in the presence of modulated losses, as well as in generalized photonic arrays where both the local resonance frequency and the loss are modulated.We showed that in the presence of second and third-nearest-neighbor hopping, the localization transition takes place at different modulation strengths for each frequency, leading to a photonic spectrum featuring a mobility edge.Focusing on a topological regime featuring zero modes, we addressed the robustness of the topological edge modes in the presence of disorder, showing that resonance frequency disorder leads to an energy splitting in the real energy, whereas loss disorder keeps states at zero energy.Finally, we address the continuum limit of the non-Hermitian model showing a similar emergence of spectral mini-bands.Our results demonstrated that pho-tonic arrays with periodically modulated losses provide a flexible platform to engineer both topological modes and criticality, making reconfigurable photonics a promising platform to explore exotic non-Hermitian states of light.

FIG. 1 .
FIG. 1.(a) Schematic of the gain-loss model with a unit cell of four sites (dashed rectangle).The local loss is modulated by the function ιvn = ιvI sin (2παn + φ), with α = 1/4 and φ = 3π/4.(b) Real part of the energy spectrum as a function of vI /t for the model shown in (a), for N = 200.The spectrum is symmetric with respect to Re(E) = 0 with an edge mode present at Re(E) = 0, which is topologically robust for the entire range of vI /t.(c)The density (DOS) of the zero mode is shown as a function of vI /t and the site index n.With an increase in the value of vI , the zero mode gets exponentially localized at the end sites.We took N = 100 for (c).

Fig.[ 1
(b)] the real part of the energy spectrum of this model for open boundary conditions as a function of v I .We can see that the edge modes with Re(E) = 0 are strongly localized at the edges, and represent the topological edge modes arising from the modulated loss.The extent of localization for the edge modes is shown in Fig.[1(c)] using the spectral function at zero real energy D 0

FIG. 2 .
FIG. 2. (a-d) Spectra as a function of φ for a chain with modulated losses, for α = 1/4 (a), α = 1/8 (b), and in the incommensurate limit α = ( √ 5−1)/2 (c), and α = ( √ 7−1)/2 (d).The energies are colored according to the spatial location of the state in the chain.It is observed that edge modes appear in spectral gaps for wide regions of φ.Panel (e) shows the spectra as a function of the wavevector of the modulation of the loss α, showing that topological edge modes appear for generic values inside the spectral gap.We took vI = 1.5 and N = 200 for (a-e).

FIG. 4 .
FIG. 4. (a) The average IPR of the chain model as a function of vR and vI , the system gets localized along the contour v 2 R + v 2 I = 4t 2 .(b,c) The inverse of localization length λ as a function of vI for vR = 0 and vR = √ 2t respectively showing the critical localization transition at vI = 2t and vI = √ 2t respectively.(d) The inverse of localization length (λ) of the chain model as a function of vR and vI , the system gets localized along the contour v 2 R + v 2 I = 4t 2 .We took α = ( √ 5 − 1)/2, N = 200, φ = 0.4π in (a-d).

5 − 1 )/ 2 .
[3(c)] and Fig.[3(d)] the energy as a function of v I for v R = √ 2t.We observe that approximately at v I = √ 2t, all the eigenstates undergo a localizationdelocalization transition.A finite value of the onsite quasiperiodicity v R , leads to a different critical value for localization transition as a function of the quasiperiodic engineered loss v I .To elucidate how the critical transition depends on both modulations, we show in Fig.[4(a)] a two-dimensional phase diagram as given by the IPR as a function of both v R and v I for α = ( √ It is observed that a localization-delocalization transition occurs following the approximate critical line v 2 R + v 2 I = 4t 2 .As a reference, the critical transition in the Hermitian AAH model corresponds to the cut v I = 0, whereas the critical transition in the purely imaginary model corresponds to the cut in v R = 0.This phenomenology highlights that the Hermitian AAH model belongs to a general family of non-Hermitian AAH models with complex modulation strength. [4(b)] the average λ versus v I for v R = 0, showing that a localization-delocalization transition occurs at v I = 2t.Similarly, setting v R as √ 2t, we see that the localization occurs at v I = √ 2t as in Fig.[4(c)].In Fig.[4(d)], we show a phase diagram according to the inverse localization length as a function of the parameters v R and v I , where we can see that the localization occurs at v 2 R + v 2 I = 4t 2 .The localization length is given as a function of the modulation strength, by the Thouless formula as 1 λ = 1 log(|v|/(2t)) , where v = v R + iv I and |v| > 2, [20].

FIG. 5 .
FIG. 5. (a-d) Real part of energy spectra for the chain model with losses as a function of the strength of modulation vI for different values of the second and third neighbor hoppings t2 and t3.The localization transition occurs for different eigenstates at different vI .The inclusion of finite t2 shows that the spectra have no particle-hole symmetry (a,b), whereas third neighbor hopping preserves particle-hole symmetry (c,d).We took N = 200, vR = 0, α = ( √ 5 − 1)/2 and φ = 0.4π in (a-d).

FIG. 6 .
FIG. 6. (a-d) Spectral density as a function of the loss disorder strength wI (a,b) and potential disorder wR (c,d), in the bulk (a,c) and at the edge (b,d).It is observed that the existence of a finite disorder decreases the bulk gap (a,c), but without destroying it.The zero edge modes are robust to the existence of disorder in loss as shown (b), whereas they develop a finite splitting in the presence of detuning disorder (d).We took N = 200 and vI = 2, and results are averaged over 100 realizations.

Fig.[ 6
(a,b)] the spectral function averaged over disorder in the loss modulation projected on the bulk (Fig.[6(a)]) and at the edge (Fig.[6(b)]).It is observed that the spectral gap in the bulk remains open in the presence of disorder (Fig.[6(a)]), and that a robust zero mode remains at the edge even at finite disorder (Fig.[6(b)] [6(c,d)]).We show in Fig.[6(c,d)] the spectral function averaged over disorder in the detuning disorder projected on the bulk (Fig.[6(c)]) and at the edge (Fig.[6(d)]).It is observed that this disorder also keeps the spectral gap open in the bulk as shown in (Fig.[6(c)]

FIG. 7 .
FIG. 7. (a) Electronic dispersion in the folded momentum space k for vR = vI = 0, and (b) unfolded dispersion in the momentum space p recover the free gas dispersion p 2 /(2m).Panels (c,d) show the spectra in the unfolded momentum p for the continuum Hermitian AAH model for two different potential strengths, showing the appearance of weakly dispersive modes and nearly free states.The insets in (c,d) show the dispersion in the folded momentum space k.Panels (e,f) show the spectra for the continuum non-Hermitian AAH model for two potential strengths.It observed that both weakly dispersive and nearly free states emerge, with the inset showing the spectra in the unfolded k space.
[8(a,b)], with Fig.[8(c,d)] showing the profiles of the Hermitian and non-Hermitian potential as a reference.In the Hermitian case shown in Fig.[8(a)

FIG. 8 .
FIG. 8. (a,b) Spectral density of the Hermitian AAH and non-Hermitian AAH model respectively is shown as a function of space x. (c,d) show the potential profile V (x) as a function of x for vI = 0 and vR nonzero and vice versa.(a) The lower energy states get localized corresponding to x when the potential is zero in (c), with an increase in the value of Re(E) the states are more extended.(b) The lower energy states get localized corresponding to x when the potential attains a ±vmax in (d).Also, with an increase in the value of Re(E), the states are more extended and at a lower value of Re(E) in comparison to the Hermitian case.